Finite Group Databasebeta

← #118132

Subgroups — #118132

Order 512.

623 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21370C211yesmaximal, characteristic
2C2×#1491C211yesmaximal, characteristic
3C2×#1487C211yesmaximal, characteristic
4#17365C211yesmaximal, characteristic
5#8077C211yesmaximal, characteristic
6#8063C211yesmaximal, characteristic
7#7863C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15#1950—22no—
16#1855—22no—
17C4×#108—22no—
18#1491—22no—
19#1491—22no—
20#1487—22no—
21#1487C414yes—
22C2×Q8⋊C8—22no—
23C2×Q8⋊C8—22no—
24#1467—22no—
25#1463—22no—
26#1461—22no—
27#1301—22no—
28#1299—22no—
29#1275—22no—
30C23×Q8Q811yescharacteristic
31C22×M16—21no—
32C22×M16—21no—
33C2×C4×Q8—21no—
34C2×C4×Q8—21no—
35C2×C4×Q8—21no—
36C2×C4×Q8—21no—
37C2×C4×Q8—21no—
38C2×C4×Q8—21no—
39C2×C4×Q8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8C2×C412yes—
44C2×C4×Q8D412yes—
45C22×C4⋊C4—21no—
46C22×C4⋊C4Q811yescharacteristic
47C22×C42—21no—
48C22×C42C2311yescharacteristic
49#432—21no—
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431C2×C412yes—
63#431D412yes—
64#431D412yes—
65C2×#108—24no—
66#423—21no—
67#423—21no—
68#423—21no—
69#423—21no—
70#423—21no—
71#423C2×C412yes—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8D412yes—
76#407—21no—
77#407D412yes—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C4×M16—21no—
81C4×M16—21no—
82C4×M16—21no—
83C4×M16—21no—
84C4×M16—21no—
85C4×M16D412yes—
86C2×#97—21no—
87C2×#97—21no—
88C2×C4×C8—21no—
89C2×C4×C8D412yes—
90#341—24no—
91Q8⋊C8—28no—
92Q8⋊C8—28no—
93#324—28no—
94C22×Q8—21no—
95C22×Q8—21no—
96C22×Q8—21no—
97C22×Q8—21no—
98C22×Q8—21no—
99C22×Q8—44no—
100C22×Q8C4⋊C412yes—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×Q811yescharacteristic
105C23×C4#4111yescharacteristic
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—44no—
110C2×M16—41no—
111C2×M16—41no—
112C2×M16—41no—
113C2×M16—82no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—41no—
117C22×C8—41no—
118C4×Q8—22no—
119C4×Q8—22no—
120C4×Q8—22no—
121C4×Q8—22no—
122C4×Q8—22no—
123C4×Q8—22no—
124C4×Q8—22no—
125C4×Q8—22no—
126C4×Q8—41no—
127C4×Q8—41no—
128C4×Q8—41no—
129C4×Q8—41no—
130C4×Q8—41no—
131C4×Q8—41no—
132C4×Q8—41no—
133C4×Q8—41no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4C4⋊C412yes—
146C2×C42—22no—
147C2×C42—22no—
148C2×C42—41no—
149C2×C42—41no—
150C2×C42—41no—
151C2×C42—41no—
152C2×C42C22×C411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42#4111yescharacteristic
160C2×C42#4111yescharacteristic
161C2×C42#4111yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164#108—48no—
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—41no—
173C4⋊C8—41no—
174C4⋊C8—41no—
175C4⋊C8—41no—
176C4⋊C8—41no—
177C22⋊C8—22no—
178C22⋊C8—22no—
179C22⋊C8—22no—
180C22⋊C8—22no—
181C22⋊C8—22no—
182C22⋊C8—41no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185#97—22no—
186#97—41no—
187#97—41no—
188#97—41no—
189#97—41no—
190#97—41no—
191C4×C8—22no—
192C4×C8—22no—
193C4×C8—22no—
194C4×C8—41no—
195C4×C8—41no—
196C4×C8C22⋊C414yes—
197C24C22⋊Q811yescharacteristic
198C2×Q8—24no—
199C2×Q8—24no—
200C2×Q8—24no—
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—44no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8—44no—
208C2×Q8—44no—
209C2×Q8—41no—
210C2×Q8—41no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—44no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—82no—
223C22×C4C2×C4⋊C411yescharacteristic
224C22×C4C4×D411yescharacteristic
225C22×C4C4×D411yescharacteristic
226C22×C4#12011yescharacteristic
227C22×C4#12011yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4C22⋊Q811yescharacteristic
232C22×C4C22⋊Q811yescharacteristic
233C22×C4C22⋊Q811yescharacteristic
234C22×C4C22⋊Q811yescharacteristic
235C22×C4C22⋊Q811yescharacteristic
236C22×C4#12311yescharacteristic
237C22×C4#12411yescharacteristic
238C22×C4#12411yescharacteristic
239C22×C4#12411yescharacteristic
240C22×C4#12611yescharacteristic
241M16—24no—
242M16—42no—
243M16—44no—
244M16—44no—
245M16—82no—
246M16—82no—
247M16—82no—
248C2×C8—24no—
249C2×C8—44no—
250C2×C8—44no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—82no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C42—42no—
268C42—42no—
269C42—42no—
270C42—42no—
271C42C2×C22⋊C411yescharacteristic
272C42#11711yescharacteristic
273C42C4×D411yescharacteristic
274C42C4×D411yescharacteristic
275C42#12011yescharacteristic
276C42#12111yescharacteristic
277C42#12111yescharacteristic
278C42#12111yescharacteristic
279C42#12111yescharacteristic
280C42#12111yescharacteristic
281C42#12311yescharacteristic
282C42#12311yescharacteristic
283C42#12311yescharacteristic
284C42#12311yescharacteristic
285C42#12611yescharacteristic
286C42#12711yescharacteristic
287C23—22no—
288C23—22no—
289C23—41no—
290C23#38511yescharacteristic
291C23#39311yescharacteristic
292C23#39611yescharacteristic
293Q8—24no—
294Q8—24no—
295Q8—42no—
296Q8—42no—
297Q8—44no—
298Q8—44no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—24no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4#38011yescharacteristic
328C2×C4#38511yescharacteristic
329C2×C4#38611yescharacteristic
330C2×C4#38811yescharacteristic
331C2×C4#38811yescharacteristic
332C2×C4#39011yescharacteristic
333C2×C4#39211yescharacteristic
334C2×C4#39411yescharacteristic
335C2×C4#39411yescharacteristic
336C2×C4#39611yescharacteristic
337C2×C4#39711yescharacteristic
338C2×C4#39911yescharacteristic
339C2×C4#47112yes—
340C2×C4#47312yes—
341C8—44no—
342C8—44no—
343C8—82no—
344C8—82no—
345C8—82no—
346C8—82no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—44no—
355C22#143911yescharacteristic
356C22#203112yes—
357C22#206412yes—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—44no—
368C4—82no—
369C4#197711yescharacteristic
370C4#197911yescharacteristic
371C4#205212yes—
372C4#209112yes—
373C2—22no—
374C2—22no—
375C2—41no—
376C2#1108511yescharacteristic
377C2#1110111yescharacteristic
378C2#1112811yescharacteristic
subgroupG/H
centerC2×C4#390
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC24C22⋊Q8
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#213702
C2×#14912
C2×#14872
#173652
#80772
#80632
#78632